Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{-5}{10(2n + 1)} \div \dfrac{-3}{5(2n + 1)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-5}{10(2n + 1)} \times \dfrac{5(2n + 1)}{-3} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ -5 \times 5(2n + 1) } { 10(2n + 1) \times -3 } $ $ p = \dfrac{-25(2n + 1)}{-30(2n + 1)} $ We can cancel the $2n + 1$ so long as $2n + 1 \neq 0$ Therefore $n \neq -\dfrac{1}{2}$ $p = \dfrac{-25 \cancel{(2n + 1})}{-30 \cancel{(2n + 1)}} = -\dfrac{25}{-30} = \dfrac{5}{6} $